View source for Astronomy:Circular orbit

{{Short description|Orbit with a fixed distance from the barycenter}}
[[Image:Newton Cannon.svg|thumb|right|[[Physics:Newton's cannonball|Isaac Newton's Cannonball]]. Path C depicts a circular orbit.]]
{{Astrodynamics}}

A '''circular orbit''' is an [[Physics:Orbit|orbit]] with a fixed distance around the [[Barycenter|barycenter]]; that is, in the shape of a [[Circle|circle]].
In this case, not only the distance, but also the speed, angular speed, [[Physics:Potential energy|potential]] and [[Physics:Kinetic energy|kinetic energy]] are constant. There is no periapsis or apoapsis. This orbit has no radial version.

Listed below is a circular orbit in astrodynamics or [[Physics:Celestial mechanics|celestial mechanics]] under standard assumptions. Here the [[Physics:Centripetal force|centripetal force]] is the gravitational force, and the axis mentioned above is the line through the [[Physics:Center of mass|center of the central mass]] [[Perpendicular|perpendicular]] to the [[Astronomy:Orbital plane|orbital plane]].

==Circular acceleration==
Transverse acceleration ([[Perpendicular|perpendicular]] to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity,  [[Physics:Circular motion|circular motion]] ensues. Taking two derivatives of the particle's coordinates with respect to time gives the [[Physics:Centripetal acceleration|centripetal acceleration]]

:<math> a\, = \frac {v^2} {r} \, = {\omega^2} {r}  </math>

where:
*<math>v\,</math> is [[Physics:Kinetic energy|orbital velocity]] of orbiting body,
*<math>r\,</math> is [[Radius|radius]] of the circle
*<math> \omega \ </math> is angular speed, measured in [[Radian|radian]]s per unit time.
The formula is [[Physics:Dimensionless quantity|dimensionless]], describing a ratio true for all units of measure applied uniformly across the formula. If the numerical value of <math> \mathbf{a}</math> is measured in meters per second per second, then the numerical values for <math>v\,</math> will be in meters per second, <math>r\,</math> in meters, and <math> \omega \ </math> in radians per second.

==Velocity==
The speed (or the magnitude of velocity) relative to the central object is constant:<ref name=lissauer2019>{{Cite book |title=Fundamental Planetary Sciences : physics, chemistry, and habitability |last1=Lissauer|first1=Jack J. |last2=de Pater|first2=Imke |year=2019 |publisher=Cambridge University Press |isbn=9781108411981 |location=New York, NY, USA |pages=604 }}</ref>{{rp|30}}
:<math> v = \sqrt{ GM\! \over{r}} = \sqrt{\mu\over{r}} </math>
where:
* <math>G</math>, is the [[Physics:Gravitational constant|gravitational constant]]
* <math>M</math>, is the [[Physics:Mass|mass]] of both orbiting bodies <math>(M_1+M_2)</math>, although in common practice, if the greater mass is significantly larger, the lesser mass is often neglected, with minimal change in the result.
*<math> \mu = GM </math>, is the [[Astronomy:Standard gravitational parameter|standard gravitational parameter]].

==Equation of motion==
The [[Astronomy:Orbit equation|orbit equation]] in polar coordinates, which in general gives ''r'' in terms of ''θ'', reduces to:{{clarify|reason=There is no θ in motion equation!|date=January 2016}}{{cn|date=August 2019}}
:<math>r={{h^2}\over{\mu}}</math>
where:
*<math>h=rv</math> is [[Astronomy:Specific angular momentum|specific angular momentum]] of the orbiting body.

This is because <math>\mu=rv^2</math>

==Angular speed and orbital period==

:<math>\omega^2 r^3=\mu</math>

Hence the [[Astronomy:Orbital period|orbital period]] (<math>T\,\!</math>) can be computed as:<ref name=lissauer2019/>{{rp|28}}
:<math>T=2\pi\sqrt{r^3\over{\mu}}</math>

Compare two proportional quantities, the [[Astronomy:Free-fall time|free-fall time]] (time to fall to a point mass from rest)

:<math>T_\text{ff}=\frac{\pi}{2\sqrt{2}}\sqrt{r^3\over{\mu}}</math> (17.7% of the orbital period in a circular orbit)

and the time to fall to a point mass in a radial parabolic orbit

:<math>T_\text{par}=\frac{\sqrt{2}}{3}\sqrt{r^3\over{\mu}}</math> (7.5% of the orbital period in a circular orbit)

The fact that the formulas only differ by a constant factor is a priori clear from [[Dimensional analysis|dimensional analysis]].{{cn|date=August 2019}}

==Energy==
[[File:Gravity Wells Potential Plus Kinetic Energy - Circle-Ellipse-Parabola-Hyperbola.png|thumb|250px|A circular orbit is depicted in the top-left quadrant of this diagram, where the [[Physics:Gravity well|gravitational potential well]] of the central mass shows potential energy, and the kinetic energy of the orbital speed is shown in red. The height of the kinetic energy remains constant throughout the constant speed circular orbit.]]

The [[Astronomy:Specific orbital energy|specific orbital energy]] (<math>\epsilon\,</math>) is negative, and
:<math>\epsilon=-{v^2\over{2}}</math>
:<math>\epsilon=-{\mu\over{2r}}</math>

Thus the [[Physics:Virial theorem|virial theorem]]<ref name=lissauer2019/>{{rp|72}} applies even without taking a time-average:{{cn|date=August 2019}}

*the kinetic energy of the system is equal to the absolute value of the total energy
*the potential energy of the system is equal to twice the total energy

The [[Astronomy:Escape velocity|escape velocity]] from any distance is {{radic|2}} times the speed in a circular orbit at that distance: the kinetic energy is twice as much, hence the total energy is zero.{{cn|date=August 2019}}

==Delta-v to reach a circular orbit==
Maneuvering into a large circular orbit, e.g. a [[Astronomy:Geostationary orbit|geostationary orbit]], requires a larger [[Astronomy:Delta-v|delta-v]] than an escape orbit, although the latter implies getting arbitrarily far away and having more energy than needed for the [[Astronomy:Orbital speed|orbital speed]] of the circular orbit. It is also a matter of maneuvering into the orbit. See also [[Astronomy:Hohmann transfer orbit|Hohmann transfer orbit]].

== Orbital velocity in general relativity ==
In [[Astronomy:Schwarzschild metric|Schwarzschild metric]], the orbital velocity for a circular orbit with radius <math>r</math> is given by the following formula:
:<math>v = \sqrt{\frac{GM}{r-r_S}}</math>
where <math>\scriptstyle r_S = \frac{2GM}{c^2}</math> is the Schwarzschild radius of the central body.

=== Derivation ===
For the sake of convenience, the derivation will be written in units in which <math>\scriptstyle c=G=1</math>.

The [[Physics:Four-velocity|four-velocity]] of a body on a circular orbit is given by:
:<math>u^\mu = (\dot{t}, 0, 0, \dot{\phi})</math>
(<math>\scriptstyle r</math> is constant on a circular orbit, and the coordinates can be chosen so that <math>\scriptstyle \theta=\frac{\pi}{2}</math>). The dot above a variable denotes derivation with respect to proper time <math>\scriptstyle \tau</math>.

For a massive particle, the components of the [[Physics:Four-velocity|four-velocity]] satisfy the following equation:
:<math>\left(1-\frac{2M}{r}\right) \dot{t}^2 - r^2 \dot{\phi}^2 = 1</math>

We use the geodesic equation:
:<math>\ddot{x}^\mu + \Gamma^\mu_{\nu\sigma}\dot{x}^\nu\dot{x}^\sigma = 0</math>
The only nontrivial equation is the one for <math>\scriptstyle \mu = r</math>. It gives:
:<math>\frac{M}{r^2}\left(1-\frac{2M}{r}\right)\dot{t}^2 - r\left(1-\frac{2M}{r}\right)\dot{\phi}^2 = 0</math>
From this, we get:
:<math>\dot{\phi}^2 = \frac{M}{r^3}\dot{t}^2</math>
Substituting this into the equation for a massive particle gives:
:<math>\left(1-\frac{2M}{r}\right) \dot{t}^2 - \frac{M}{r} \dot{t}^2 = 1</math>
Hence:
:<math>\dot{t}^2 = \frac{r}{r-3M}</math>

Assume we have an observer at radius <math>\scriptstyle r</math>, who is not moving with respect to the central body, that is, their [[Physics:Four-velocity|four-velocity]] is proportional to the vector <math>\scriptstyle \partial_t</math>. The normalization condition implies that it is equal to:
:<math>v^\mu = \left(\sqrt{\frac{r}{r-2M}},0,0,0\right)</math>
The [[Dot product|dot product]] of the [[Physics:Four-velocity|four-velocities]] of the observer and the orbiting body equals the gamma factor for the orbiting body relative to the observer, hence:
:<math>\gamma = g_{\mu\nu}u^\mu v^\nu = \left(1-\frac{2M}{r}\right) \sqrt{\frac{r}{r-3M}} \sqrt{\frac{r}{r-2M}} = \sqrt{\frac{r-2M}{r-3M}}</math>
This gives the [[Physics:Kinetic energy|velocity]]:
:<math>v = \sqrt{\frac{M}{r-2M}}</math>
Or, in SI units:
:<math>v = \sqrt{\frac{GM}{r-r_S}}</math>

[[File:counterintuitive_orbital_mechanics.svg|thumb|250px|At the top of the diagram, a satellite in a clockwise circular orbit (yellow spot) launches objects of negligible mass:<br />(1 - blue) towards Earth,<br />(2 - red) away from Earth,<br />(3 - grey) in the direction of travel, and<br />(4 - black) backwards of the direction of travel.<br /><br />Dashed ellipses are orbits relative to Earth. Solid curves are perturbations relative to the satellite: in one orbit, (1) and (2) return to the satellite having made a clockwise loop on either side of the satellite. Unintuitively, (3) spirals farther and farther behind whereas (4) spirals ahead.]]

==See also==
*[[Astronomy:Elliptic orbit|Elliptic orbit]]
*[[Astronomy:List of orbits|List of orbits]]
*[[Two-body problem]]

==References==
{{Reflist}}

{{Orbits|state=expanded}}

{{DEFAULTSORT:Circular Orbit}}
[[Category:Orbits]]

{{Sourceattribution|Circular orbit}}